Inductive Definition and Domain Theoretic Properties of Fully Abstract

Abstract

A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF +"parallel conditional function"), respectively, is presented. It is based ongeneral notions of sequential computational strategies and wittingly consistentnon-deterministic strategies introduced by the author in the seventies.Although these notions of strategies are old, the definition of the fullyabstract models is new, in that it is given level-by-level in the finite typehierarchy. To prove full abstraction and non-dcpo domain theoretic propertiesof these models, a theory of computational strategies is developed. This isalso an alternative and, in a sense, an analogue to the later game strategysemantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong;and Nickau. In both cases of PCF and PCF^+ there are definable universal(surjective) functionals from numerical functions to any given type,respectively, which also makes each of these models unique up to isomorphism.Although such models are non-omega-complete and therefore not continuous in thetraditional terminology, they are also proved to be sequentially complete (aweakened form of omega-completeness), "naturally" continuous (with respect toexisting directed "pointwise", or "natural" lubs) and also "naturally"omega-algebraic and "naturally" bounded complete -- appropriate generalisationof the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page